Solving nonlinear differential equations on a quantum computer

Title: Quantum algorithms for nonlinear differential equations

Authors: Seth Lloyd, Giacomo De Palma, Can Gokler, Bobak Kiani, Zi-Wen Liu, Milad Marvian, Felix Tennie, Tim Palmer

First Author’s Institution: MIT

Status: Pre-print on arXiv

Quantum computers will undoubtedly have a transformative impact on society once they’re scaled up to the order of millions of qubits and can reliably perform error correction. We often hear about exciting applications like simulating complex quantum systems, factoring large numbers, and speeding up numerical optimization, but the potential advantages go well beyond these famous examples! For instance, a 2014 paper by Dominic Berry showed that quantum computers can solve linear ordinary differential equations (ODEs) exponentially faster than the best known classical algorithm with respect to the size of the input. 

In November 2020, two papers appeared on arXiv within a week of each other demonstrating a similar exponential improvement for non-linear ODEs. We’ll focus on the paper by Lloyd et al., but I encourage you to check out the similar paper by Liu et al. Both provide algorithms whose cost scales polylogarithmically (that is, as O(log^k(n))), whereas the fastest classical algorithms all scale polynomially (O(n^k)). There is a subtlety involved, though, since the solution to the ODE is encoded in the amplitudes of a quantum state as in the linear systems algorithm by Harrow, Hassidim, and Lloyd (HHL), but there’s still potential for significantly speeding up lots of practical problems. In fact, a large part of scientific computing is just based on solving different kinds of ODEs! 

Formulating the Problem

First, let’s lay out the problem to be solved. Suppose you have an initial state which is a complex vector with d components encoded as the state of a quantum register with log(d) qubits. We want to solve a differential equation of the form: 

\dot{x} + f( x, x^{\dagger} ) x = b(t)

Where f is a d\times d matrix and polynomial function of x and its Hermitian conjugate. By solve, we mean prepare the quantum state corresponding to the solution of the above equation at any time T given the initial state as input.

We should clarify what we mean by encoding an initial state in a quantum register. Consider a normalized linear combination of basis vectors x = \sum_i c_i x_i. The corresponding quantum state is

\left|x\right> = \sum_i c_i\left|i\right>,

where \left|i\right> is the i’th computational basis state. As usual, the quantum state requires only log(d) qubits to represent the d basis vectors. Notice, however, that if we wanted to obtain a classical representation of x from \left|x\right>, we would need to conduct O(d) measurements of the quantum state. This is a major caveat to the non-linear ODE algorithm; although the algorithm itself runs in polylog(d) time, extracting the full solution takes poly(d) time. Nevertheless, you can extract some useful information like power spectra from such a state efficiently, maintaining the exponential advantage.

The Algorithm

First, let’s look at the case where there’s no driving term b(t) and f(x,x^{\dagger}) is anti-Hermitian, that is, i times some Hermitian operator. Then the ODE looks just like the Schrodinger equation but with a nonlinear Hamiltonian. It turns out there are already methods for implementing such an equation! Succinctly, by performing ordinary time-evolution on many copies of the same quantum state using a carefully chosen Hamiltonian, you can get one of those copies to obey the nonlinear Schrodinger equation (for short times) after tracing out the other degrees of freedom. For an introduction to the concept of ‘tracing out’ degrees of freedom, see Wikipedia: partial trace.

Previous quantum methods for solving linear ODEs typically use the HHL algorithm for solving linear systems of equations (this paper deserves its own qubyte – let us know in the comments if that’s something you’d like to see!). In this paper, the authors combine HHL with the nonlinear Schrodinger equation approach to solve nonlinear ODEs in the presence of a driving term even when f is not anti-Hermitian.

Explicitly, they take the equation for the nonlinear Schrodinger equation corresponding to the desired ODE, map it to a problem in linear algebra using standard techniques from numerical analysis, and solve the resulting system with HHL. The output is given in the form of a quantum history state, meaning that the algorithm returns a quantum state composed of a solution vector and a time register

\left|X\right> = \sum_t \left|x_t\right>\left|t\right>

where \left|x_t\right> is the solution to the ODE at timestep t.

Ultimately, the speedup comes from the HHL subroutine, which is exponentially faster than the best classical algorithm for solving linear systems with respect to the size of the system. The error in the quantum nonlinear ODE solver is dominated by the discretization error from representing the ODE as a linear system of equations. Naturally, the performance of the algorithm will depend strongly on properties of the ODE to be solved and the numerical method used to discretize the system.

To achieve precision \varepsilon in the output for total simulated time T and a timestep of \Deltat, the algorithm requires

|E|^2T\Delta t < O(\varepsilon),

where |E|^2 is the average norm squared of the eigenvalues of the ‘non-Hermitian Hamiltonian’ f over the total simulation time – this is reasonable provided either that the eigenvalues don’t grow exponentially fast or we solve for sufficiently short times; essentially this means the algorithm works well for systems that aren’t chaotic. Again, this doesn’t necessarily mean that quantum computers will replace classical computers for solving non-linear ODEs in all cases, but it may well open the door to some exciting applications down the line!